Which value, to the nearest tenth, is the smallest solution of f(x) = g(x) if f(x) = ln(x + 5) – 1 and g(x) = x3 – 2x + 1?

Answer:

There are two possibilities of determine which solution is the smallest one:

(i) The smallest solution is the value of [tex]x[/tex] has the smallest distance with respect to origin: [tex](x_{2},y_{2}) = (0.18,0)[/tex]

(ii) The smallest solution is the most negative value: [tex](x_{1},y_{1}) = (-1.577,0)[/tex]

Step-by-step explanation:

Let [tex]f(x) = g(x)[/tex], [tex]f(x) = \ln (x+5)-1[/tex] and [tex]g(x) = x^{3}-2\cdot x + 1[/tex], then we have the following implicit equation:

[tex]\ln (x+5)-x^{3}+2\cdot x -2 = 0[/tex] (1)

This equation cannot be solved by analytical means. There are two different strategies to solve this expression: (i) Numerical methods, (ii) Graphical methods. We decide to use the second method:

Let suppose that we use this function [tex]h(x) = \ln (x+5) -x^{3}+2\cdot x - 2[/tex]. By the help of a graphing tool, we find the smallest solution such that [tex]h(x) = 0[/tex]. There are three possible solutions: [tex](x_{1},y_{1}) = (-1.577,0)[/tex], [tex](x_{2},y_{2}) = (0.18,0)[/tex], [tex](x_{3}, y_{3}) = (1.376, 0)[/tex]

There are two possibilities of determine which solution is the smallest one:

(i) The smallest solution is the value of [tex]x[/tex] has the smallest distance with respect to origin: [tex](x_{2},y_{2}) = (0.18,0)[/tex]

(ii) The smallest solution is the most negative value: [tex](x_{1},y_{1}) = (-1.577,0)[/tex]

The smallest smallest solution of f(x) = g(x) will be [tex]\rm \left(-1.577,0.223\right)[/tex]

A function is a statement, rule, or law that specifies the connection between two variables. Functions are common in mathematics and are required for the formulation of physical connections.

Given functions;

f(x) = ln(x + 5) – 1

g(x) = x³ – 2x + 1

After graphing both the function, we get the three solutions of the given equation,

[tex]\rm \left(-1.577,0.223\right)[/tex]

[tex]\left(0.18,0.645\right)[/tex]

[tex]\left(1.376,0.853\right)[/tex]

Hence,the smallest smallest solution of f(x) = g(x) will be [tex]\rm \left(-1.577,0.223\right)[/tex]

To learn more about the function, refer to:

https://brainly.com/question/5245372

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